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# Stress-Strain Theory - PowerPoint PPT Presentation

Stress-Strain Theory. Under action of applied forces, solid bodies undergo deformation, i.e., they change shape and volume. The static mechanics of this deformations forms the theory of elasticity, and dynamic mechanics forms elastodynamic theory. . u(x+dx). dx. dx’. dx. dx’. u(x). x. x’.

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Under action of applied forces, solid bodies undergo deformation, i.e., they change shape and volume. The static mechanics of this deformations forms the theory of elasticity, and dynamic mechanics forms elastodynamic theory.

dx

dx’

dx

dx’

u(x)

x

x’

Length squared: dl = dx + dx + dx = dx dx

2

2

2

2

i

i

1

2

3

dl = dx’ dx’ = (du +dx )

2

2

i

i

i

i

= du du + dx dx + 2 dudx

i

i

i

i

i

i

Strain Tensor

After deformation

Displacement vector: u(x) = x’- x

dx

dx’

dx

dx’

u(x)

x

x’

Length squared: dl = dx + dx + dx = dx dx

2

2

2

2

i

i

1

2

3

dl = dx’ dx’ = (du +dx )

2

2

i

i

i

i

Length change:dl - dl = du du + 2du dx

2

2

= du du + dx dx + 2dudx

i

i

i

i

i

i

i

i

i

i

du = du dx

Substitute

i

i

j

dx

j

Strain Tensor

After deformation

(1)

into equation (1)

dx

dx’

dx

dx’

u(x)

x

x’

(1)

into equation (1)

Length change:dl - dl = U U

2

2

Strain Tensor

Length change:dl - dl = du du + 2du dx

i

i

2

2

i

i

i

i

(du + du + du du )dx dx

du = du dx

=

i

j

j

k

k

i

Substitute

dx

dx

dx

dx

i

i

j

dx

j

j

i

i

j

Strain Tensor

After deformation

(2)

Problem

V > C

Problem

V > C

V < C

Elastic Strain Theory

Elastodynamics

e

=

=

=

Length Change

L L

xx

Length

L’

L’

L

L

Acoustics

e

=

=

=

Length Change

L L

xx

Length

L’

L’

L

L

Acoustics

e

=

=

=

Length Change

L L

xx

Length

L’

L’

L

L

Acoustics

No Shear Resistance = No Shear Strength

dw

du

dz

Acoustics

dw, du << dx, dz

dx

AreaChange

(dz+dw)(dx+du)-dxdz

=

=

+ O(dudw)

dx dz

dx dz

Area

dw du

=

+

dz

dx

e

+

dw

=

du

xx

U

=

dz

e

zz

Acoustics

really small

big +small

big +small

Infinitrsimal strain

assumption: e<.00001

Dilitation

dx

e

-

P =

(

)

+

Bulk Modulus

xx

U

=

e

dx

zz

1D Hooke’s Law

pressure

strain

-k

du

Infinitrsimal strain

assumption: e<.00001

F/A =

Pressure is F/A of outside

media acting on face of box

F/A =

(

)

+

xx

Compressional

Source or Sink

k

Larger = Stiffer Rock

=

e

e

zz

zz

Hooke’s Law

Dilation

e

k

U

Infinitrsimal strain

assumption: e<.00001

e

k

-

P =

(

)

+ S

+

xx

Bulk Modulus

..

..

-

-

dP

dP

r

r

;

w =

u =

dx

dz

Net force = [P(x,+dx,z,t)-P(x,z,t)]dz

density

x,

..

k

Larger = Stiffer Rock

r

u

P (x,z,t)

P (x+dx,z,t)

ma = F

-dxdz

1st-Order Acoustic Wave Equation

..

P

-

r

u =

u=(u,v,w)

..

..

-

-

dP

dP

r

r

;

w =

u =

dx

dz

density

k

Larger = Stiffer Rock

P (x,z,t)

P (x+dx,z,t)

1st-Order Acoustic Wave Equation

..

P

-

r

u =

(1)

(3)

(4)

..

..

k

P

= -

U

(2)

..

]

P

-

[

u =

1

r

..

k

]

P

-

[

P =

1

r

(Newton’s Law)

(Hooke’s Law)

Divide (1) by density and take Divergence:

Take double time deriv. of (2) & substitute (2) into (3)

2nd-Order Acoustic Wave Equation

..

k

]

P

-

[

P =

1

r

..

k

P

P =

r

k

c =

2

Substitute velocity

r

..

2

P

c

P =

2

Constant density assumption

..

..

P

-

k

r

k

]

u =

P

-

[

1. Hooke’s Law: P

P =

= -

U

1

r

2. Newton’s Law:

3. Acoustic Wave Eqn:

k

c =

2

r

..

2

P

c

;

P =

2

Constant density assumption

Body Force Term

+ F

1. Utah and California movingE-W apart at 1cm/year.

Calculate strain rate, where distance is 3000 km. Is it e or e ?

2. LA. coast andSacremento moving N-S apart at 10cm/year.

Calculate strain rate, where distance is 2000 km. Is is e or e ?

xx

xx

xy

xy

3. A plane wave soln to W.E. is u= cos

(kx-wt) i.

Compute divergence. Does the volume change

as a function of time? Draw state of deformation boxes

Along path

U

n

dl

U

= lim

k

e

-

A

P =

(

)

+

A 0

xx

+ U(x,z+dz)cos(90)dx

+ U(x,z+dz)cos(90)dx

- U(x,z)dz

dxdz

dxdz

dxdz

dxdz

(x+dx,z+dz)

n

n

e

zz

Divergence

= U(x+dx,z)dz

>> 0

= 0

No sources/sinks inside box.

What goes in must come out

Sources/sinks inside box.

What goes in might not come out

U(x,z)

U(x+dx,z)

(x,z)